This is the kind of problem I find very troublesome. First, MadMatt is right. The problem is not the angle of the tuners. (I wouldn't bother turning them. But, that's me.) You have two problems.
You say the instrument doesn't stay in tune. You must first of all, be patient. The strings may take a week or two to finish their initial stretching. Then, if you tune it and it holds pretty well while you play, say half an hour or so, maybe you're expecting too much. But, if you adjust a tuner and the string immediately falls in pitch the problem is most probably the tuner itself. If this is the problem, you may need to replace the tuners. You can find exact replacements and switch them out very easily, or have a luthier replace them with higher quality machines.
Your second problem is the intonation of the instrument. You say you can't get it in tune with the tuner and itself. I'm an intonation geek and this drives me crazy. All fretted instruments have this to some degree, but some are so close no one can tell. On the other end of the spectrum, some are so bad they are unplayable. We can't know where yours is on this scale. If you tune it, turn off the tuner (don't let the electronic gadget make you over think), and play it, how does it sound to you? If it's all right, enjoy. If you can't take it, the intonation is not good enough for you. If it is poorly set up, some of this can be remedied by correcting string height, fret leveling, and so on. This should be done by someone with experience (professional luthier). A professional may also be able to estimate how correctable this will be before you pay to have it done. A hobbyist will not. Set up may solve the issue for you. If not, the instrument's intonation may not be capable of making you happy.
Please, let us know how this turns out for you. And good luck.